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On a new normalization for tractor covariant derivatives. (English) Zbl 1264.58029
A parabolic geometry is a quotient $$G/P$$ of a real semisimple Lie group by the action of a parabolic subgroup $$P$$. More generally, a parabolic geometry of type $$(G,P)$$ on a manifold $$M$$ is a pair consisting of a principal $$P$$-bundle $$\mathcal{G} \to M$$ and a Cartan connection $$\omega \in \Omega^1 ( \mathcal{G} , \mathfrak{g})$$. Examples of parabolic geometries on a manifold include CR, conformal or projective structures.
If $$(\mathcal{G} , \omega)$$ is a (regular and normal) parabolic geometry on a manifold $$M$$, and $$\mathbb{V}$$ is a $$G$$-module, the vector bundle $$V \to M$$ associated to $$\mathcal{G}$$ and the representation $$\mathbb V$$ is called the tractor bundle. The normal Cartan connection $$\omega$$ on $$\mathcal{G}$$ induces a covariant derivative $$\nabla^\omega$$ on $$V$$.
As it is carefully explained in the detailed introduction of the paper, there exists a curved version of the Berstein-Gelfand-Gelfand (BGG) resolution of an irreducible $$G$$-module $$\mathbb V$$. This resolution is made up of invariant differential operators $$D_i$$. The first of these operators, $$D_0$$, is overdetermined and, in the flat case $$M = G/P$$, the tractor covariant derivative $$\nabla^\omega$$ is already known to yield the prolongation of $$D_0$$.
The main result of this paper consists of the curved analogue for such a prolongation. To prove this result, the authors develop a new normalization of $$\nabla^\omega$$, which is invariantly defined and generalizes some other constructions existing in the literature.
In Section 4, the authors extend this construction to the other operators $$D_i$$ in the BGG sequence. These normalizations require more complicated modifications (with differential terms) of the exterior covariant derivative $$d^{\nabla^\omega}$$. As the authors mention, their approach “is based on standard BGG techniques”.
Finally, the paper includes some examples, taken from projective, conformal and Grassmannian geometry, that illustrate the results commented above.
This paper is very clearly written, in an elegant and coordinate independent manner.

##### MSC:
 58J70 Invariance and symmetry properties for PDEs on manifolds 53A30 Conformal differential geometry (MSC2010) 53A20 Projective differential geometry 58A32 Natural bundles 53A55 Differential invariants (local theory), geometric objects
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##### References:
 [1] Bailey, T. N., Eastwood, M. G., Gover, A. R.: Thomas’s structure bundle for confor- mal, projective and related structures. Rocky Mountain J. Math. 24, 1191-1217 (1994) · Zbl 0828.53012 [2] Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing Spinors on Riemannian Manifolds. Seminarberichte 108, Humboldt Universität, Sektion Mathematik, Berlin (1990) · Zbl 0705.53004 [3] Branson, T.: Conformal structure and spin geometry. In: Dirac Operators: Yesterday and Yoday, Int. Press, Sommerville, MA, 163-191 (2005) · Zbl 1109.53051 [4] Branson, T., \check Cap, A., Eastwood, M. G., Gover, A. R.: Prolongations of geometric overdeter- mined systems. Int. J. Math. 17, 641-664 (2006) · Zbl 1101.35060 [5] Calderbank, D., Diemer, T.: Differential invariants and curved Bernstein-Gelfand-Gelfand sequences. J. Reine Angew. Math. 537, 67-103 (2001) · Zbl 0985.58002 [6] \check Cap, A.: Infinitesimal automorphisms and deformations of parabolic geometries. J. Eur. Math. Soc. 10, 415-437 (2008) · Zbl 1161.32020 [7] \check Cap, A., Slovák, J.: Weyl structures for parabolic geometries. Math. Scand. 93, 53-90 (2003) · Zbl 1076.53029 [8] \check Cap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory. Math. Surveys Monogr. 154, Amer. Math. Soc. (2009) · Zbl 1183.53002 [9] \check Cap, A., Slovák, J., Sou\check cek, V.: Bernstein-Gelfand-Gelfand sequences. Ann. of Math. 154, 97-113 (2001) · Zbl 1159.58309 [10] Cartan, É.: Les espaces ‘a connexion conforme. Ann. Soc. Polon. Math. 2, 171-221 (1923) · JFM 50.0493.01 [11] Cartan, É.: Sur les variétés ‘a connexion projective. Bull. Soc. Math. France 52, 205-241 (1924) · JFM 50.0500.02 [12] Chern, S. S., Moser, J. K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219- 271 (1974) · Zbl 0302.32015 [13] Dunajski, M., Tod, P.: Four dimensional metrics conformal to Kähler. Math. Proc. Cambridge Philos. Soc. 148, 485-503 (2010) · Zbl 1188.53078 [14] Eastwood, M. G.: Notes on conformal differential geometry. Suppl. Rend. Circ. Mat. Palermo 43, 57-76 (1996) · Zbl 0911.53020 [15] Eastwood, M. G.: Notes on projective differential geometry. In: Symmetries and Overdeter- mined Systems of Partial Differential Equations, IMA Vol. Math. Appl. 144, Springer, New York, 41-60 (2008) · Zbl 1186.53020 [16] Eastwood, M. G., Gover, A. R.: Prolongation on contact manifolds. arXiv:0910.5519 · Zbl 1251.58007 [17] Eastwood, M. G., Matveev, V.: Metric connections in projective differential geometry. In: Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl. 144, Springer, New York, 339-351 (2008) · Zbl 1144.53027 [18] Gover, A. R.: Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature. J. Geom. Phys. 60, 182-204 (2010) · Zbl 1194.53038 [19] Gover, A. R., \check Silhan, J.: The conformal Killing equation on forms-prolongations and ap- plications, Differential Geom. Appl. 26, 244-266 (2008) · Zbl 1144.53036 [20] Gover, A. R., Slovák, J.: Invariant local twistor calculus for quaternionic structures and re- lated geometries. J. Geom. Phys. 32, 14-56 (1999) · Zbl 0981.53031 [21] Gover, A. R., Somberg, P., Sou\check cek, V.: Yang-Mills detour complexes and conformal geom- etry. Comm. Math. Phys. 278, 307-327 (2008) · Zbl 1141.58013 [22] Hammerl, M.: Invariant prolongation of BGG-operators in conformal geometry. Arch. Math. (Brno) 44, 367-384 (2008) · Zbl 1212.53014 [23] Hammerl, M.: Natural prolongations of BGG operators. Thesis, Univ. of Vienna (2009) [24] Hammerl, M., Somberg, P., Sou\check cek, V., \check Silhan, J.: Invariant prolongation of overdetermined PDEs in projective, conformal, and Grassmannian geometry. Ann. Global Anal. Geom. 42, 121-145 (2012) · Zbl 1270.53024 [25] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. (2) 74, 329-387 (1961) · Zbl 0134.03501 [26] Leitner, F.: Conformal Killing forms with normalisation condition. Suppl. Rend. Circ. Mat. Palermo (2) 75, 279-292 (2005) · Zbl 1101.53040 [27] Morimoto, T.: Lie algebras, geometric structures and differential equations on filtered mani- folds. In: Lie Groups, Geometric Structures and Differential Equations-One Hundred Years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math. 37, Math. Soc. Japan, Tokyo, 205-252 (2002) · Zbl 1048.58015 [28] Neusser, K.: Prolongation on regular infinitesimal flag manifolds. Int. J. Math. 23, no. 4, art. ID 1250007, 41 pp. (2012) · Zbl 1256.35029 [29] Penrose, R., Rindler, W.: Spinors and Space-Time Vols. 1, 2, Cambridge Univ. Press (1984, 1986) · Zbl 0538.53024 [30] Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503- 527 (2003) · Zbl 1061.53033 [31] Sharpe, R. W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Grad. Texts in Math. 166, Springer (1997) · Zbl 0876.53001 [32] Spencer, D. C.: Overdetermined systems of linear partial differential equations. Bull. Amer. Math. Soc. 75, 179-239 (1969) · Zbl 0185.33801
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